In addition, the absorption rate of the QSD \(\nu _\text{QS}\)  is \(\theta _0\) , i.e. \(\mathbb {P}_{\nu _\text{QS}}(\tau _\partial \ge n)=\theta _0^n\) 
(see again [1]}), and \(\nu _\text{QS}(\eta )=1\) . It is also easy to see that the measure \(\nu _\text{QS}\)  is the unique
QSD satisfying \(\nu _\text{QS}(\eta )>0\)  and \(\nu _\text{QS}(V)<+\infty \)  and that the function \(\eta \)  is an eigenfunction of the semigroup, since
\(P_n\eta =\theta _0^n\eta \)  for all \(n\ge 0\) . After giving the setting of our results on periodic processes and basic properties of
their QSDs in Section , the quasi-limiting properties recalled above are extended to the periodic case in
Section  below. Of course, exponential convergence of the conditional distributions does not hold in general because of
the periodicity of the process [2]}, [3]}, [4]}, [5]}, yet we are able to
obtain periodic exponential estimates in total variation.
